So I'm gonna make a to try to get into using magic in Skyrim.
However, she is likely to use her arms more than her spells. (Which is mainly frost magic.)

Yeah my idea is to make a frost centered battlemage, with typical frost spells and rarely fire spells.
But would she join the companions? I mentioned that she especially uses her weapon but I'm not sure if the companions want a magic user, they don't seem to trust such things.

They may make milk dinker jokes behind your back but as long you use your arm in combat most of the time then I am sure they will look past that.